Chapter 3

3 The Mathematics of Sharing

3.1 Fair-Division Games3.2 Two Players: The Divider-ChooserMethod3.3 The Lone-Divider Method3.4 The Lone-Chooser Method3.5 The Last-Diminsher Method

3.6 The Method of Sealed Bids

3.7 The Method of MarkersCopyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 2

The Method of Markers

The method of markers (discrete fair-divisionmethod).Requires:(1)there are many more items to be dividedthan there are players in the game and(2)the items are reasonably close in value.

Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 3

The Method of Markers

- items lined up in a random but fixedsequence (array).- Each of the players make an bid on theitems.- A players bid consists of dividing the arrayinto segments of consecutive items (asmany segments as there are players) sothat each of the segments represents a fairshare of the entire set of items.Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 4

The Method of Markers

For convenience, we might think of the arrayas a string. Each player then cuts the stringinto N segments, each of which he or sheconsiders an acceptable share. (Notice thatto cut a string into N sections, we need N 1cuts.) In practice, one way to make the cutsis to lay markers in the places where the cutsare made. Thus, each player can make his orher bids by placing markers so that theydivide the array into N segments.Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 5

The Method of Markers

To ensure privacy, no player should see themarkers of another player before laying downhis or her own.The final step is to give to each player one ofthe segments in his or her bid.

Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 6

Example 3.11 Dividing the Halloween

LeftoversAlice, Bianca, Carla, and Dana want to dividethe Halloween leftovers shown in Fig. 3-16among themselves. There are 20 pieces, buthaving each randomly choose 5 pieces is notlikely to work wellthe pieces aretoo varied for that. Their teacher,Mrs. Jones, offers to divide thecandy for them, but the childrenreply that they just learned abouta cool fair-division game theywant to try, and they can do itthemselves, thank you.Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 7

Example 3.11 Dividing the Halloween

LeftoversArrange the 20 pieces randomly in an array.

Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 8

Example 3.11 Dividing the Halloween

LeftoversStep 1 (Bidding)Each child writes down independently on apiece of paper exactly where she wants toplace her three markers.The A-labels indicate the position of Alicesmarkers (A1 denotes her first marker, A2 hersecond marker, and A3 her third and lastmarker).Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 9

Example 3.11 Dividing the Halloween

LeftoversStep 1 (Bidding)

Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 10

Example 3.11 Dividing the Halloween

LeftoversStep 1 (Bidding)Alices bid means that she is willing to acceptone of the following as a fair share of thecandy:(1)pieces 1 through 5 (first segment),(2)pieces 6 through 11 (second segment),(3)pieces 12 through 16 (third segment), or(4)pieces 17 through 20 (last segment).

Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 11

Example 3.11 Dividing the Halloween

LeftoversStep 2 (Allocations)This is the tricky part, where we are going togive to each child one of the segments in herbid. Scan the array from left to right until thefirst first marker comes up.

Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 12

Example 3.11 Dividing the Halloween

LeftoversStep 2 (Allocations)This means that Bianca will be the first playerto get her fair share consisting of the firstsegment in her bid (pieces 1 through 4).

Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 13

Example 3.11 Dividing the Halloween

LeftoversStep 2 (Allocations)Bianca is done now, all her markers can beremoved.Continue scanning from left to right lookingfor the first second marker. Here the firstsecond marker isCarlas C2, soCarla will be thesecond playertaken care of.Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 14

Example 3.11 Dividing the Halloween

LeftoversStep 2 (Allocations)Carla gets the second segment in her bid(pieces 7 through 9). Carlas remainingmarkers can now be removed.

Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 15

Example 3.11 Dividing the Halloween

LeftoversStep 2 (Allocations)Continue scanning from left to right lookingfor the first third marker. Here there is a tiebetween Alices A3 and Danas D3.

Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 16

Example 3.11 Dividing the Halloween

LeftoversStep 2 (Allocations)As usual, a coin toss is used to break the tieand Alice will be the third player to goshewill get the third segment in her bid

Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 17

Example 3.11 Dividing the Halloween

LeftoversStep 2 (Allocations)Dana is the last player and gets the lastsegment in her bid

Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 18

Example 3.11 Dividing the Halloween

LeftoversStep 2 (Allocations)At this point each player has gotten a fairshare of the 20 pieces of candy. The amazingpart is that there is leftover candy!

Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 19

Example 3.11 Dividing the Halloween

LeftoversStep 3 (Dividing the Surplus)The easiest way to divide the surplus is torandomly draw lots and let the players taketurns choosing one piece at a time until thereare no more pieces left. Here the leftoverpieces are 5, 6, 10, and 11 The players nowdraw lots; Carla gets to choose first and takespiece 11. Dana chooses next and takes piece5. Bianca and Alicereceive pieces 6 and 10,respectively.Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 20

The Method of Markers Generalized

The ideas behind Example 3.11 can be easilygeneralized to any number of players. Wenow give the general description of themethod of markers with N players and Mdiscrete items.

PreliminariesThe items are arranged randomly into anarray. For convenience, label the items 1through M, going from left to right.Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 21

The Method of Markers Generalized

Step 1 (Bidding)Each player independently divides the arrayinto N segments (segments 1, 2, . . . , N) byplacing N 1 markers along the array. Thesesegments are assumed to represent the fairshares of the array in the opinion of thatplayer.

Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 22

The Method of Markers Generalized

Step 2 (Allocations)Scan the array from left to right until the firstfirst marker is located. The player owning thatmarker (lets call him P1) goes first and getsthe first segment in his bid. (In case of a tie,break the tie randomly.) P1s markers areremoved, and we continue scanning from leftto right, looking for the first second marker.

Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 23

The Method of Markers Generalized

Step 2 (Allocations)The player owning that marker (lets call herP2) goes second and gets the secondsegment in her bid. Continue this process,assigning to each player in turn one of thesegments in her bid. The last player gets thelast segment in her bid.Step 3 (Dividing the Surplus)The players get to go in some random orderand pick one item at a time until all thesurplus items are given out.Copyright 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 3.1 - 24

The Method of Markers: Limitation

Despite its simple elegance, the method ofmarkers can be used only under some fairlyrestrictive conditions: it assumes that everyplayer is able to divide the array of items intosegments in such a way that each of thesegments has approximately equal value. Thisis usually possible when the items are of smalland homogeneous value, but almostimpossible to accomplish when there is acombination of expensive and inexpensiveitems.